for an atom initially in a definite state (ground or excited) entering a cavity containing a known number of photons, then the state of the atom-cavity system at later times becomes a superposition of the new eigenstates of the atom-cavity system: This change in eigenstates due to the alteration of the Hamiltonian caused by the atom-field interaction is sometimes called "dressing" the atom, and the new eigenstates are referred to as the dressed states. The energy difference between the dressed states is: Of particular interest is the case where the cavity frequency is perfectly resonant with the transition frequency of the atom, so . In the resonant case, the dressed states are: With energy difference . Thus the interaction of the atom with the field splits the degeneracy of the states and by . This non-linear hierarchy of energy levels scaling as is known as the Jaynes-Cummings ladder. This non-linear splitting effect is purely quantum mechanical, and cannot be explained by any semi-classical model. Collapse and revival of probabilities Consider an atom initially in the ground state interacting with a field mode initially prepared in a coherent state, so the initial state of the atom-field system is: For simplicity, take the resonant case (), then the Hamiltonian for the nth number subspace is: Using this, the time evolution of the atom-field system will be: Note neither of the constant factors nor contribute to the dynamics beyond an overall phase, since they represent the zero-point energy. In this case, the probability to find the atom having flipped to the excited state at a later time is: Where we have identified to be the mean photon number in a coherent state. If the mean photon number is large, then since the statistics of the coherent state are Poissonian we have that the variance-to-mean ratio is . Using this result and expanding around to lowest non-vanishing order in gives: Inserting this into the sum yields a complicated product of exponentials: For "small" times such that , the inner exponential inside the double exponential in the last term can be expanded up second order to obtain: This result shows that the probability of occupation of the excited state oscillates with effective frequency . It also shows that it should decay over characteristic time: The collapse can be easily understood as a consequence of destructive interference between the different frequency components as they de-phase and